Formal diagram coefficient-to-observable adoption — Examples

Use the massless $m=M=0$ case of the formal tree theorem Formal tree scattering and scalar-decay coefficients, including its stated ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis, with $s>0$. Explicitly take incoming $p_1,p_2$ and outgoing $k_1,k_2$ on the positive massless shell, so each $(E,\mathbf{q})$ has $E=\mathrm{\mid }\mathbf{q}\mathrm{\mid }>0$, with $p_1+p_2=k_1+k_2=P$ and $h(P,P)=s>0$. Use the theorem's proper orthochronous center-of-momentum chart $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$; $\theta$ is the Euclidean angle between the nonzero spatial parts of $\mathrm{\Lambda }{p}_1$ and $\mathrm{\Lambda }{k}_1$. The differential coefficient at $\theta =0$ is twice its value at $\theta =\pi \mathrm{/}2$; integrating over the full sphere gives ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{4\pi {\alpha }^2(\mathrm{\hslash }c{)}^2}{3s}\mathrm{.}$ All three quantities are formal coefficients in the adopted map, not event counts or exact measured cross sections.

The angular contrast is a simple consequence of the massless tree angular polynomial.

Assume Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as required by the cited phase-space map and the Borel restriction of Lebesgue measure.

Let $s>0$, dimensionless $\lambda \in \mathbb{R}$, parent energy ${\mathsf{M}}_0>0$, and $G_E\in \mathbb{C}\setminus \{0\}$ in joules. For scattering, take the incoming $p_1,p_2$ and outgoing $k_1,k_2$ on the positive massless scalar shell, each $(E,\mathbf{q})$ having $E=\mathrm{\mid }\mathbf{q}\mathrm{\mid }>0$, with common total $p_1+p_2=k_1+k_2=P$, $h(P,P)=s>0$, and evaluate the coefficient in a proper orthochronous center-of-momentum chart $\mathrm{\Lambda }P=(\sqrt{s},\mathbf{0})$ as in the phase-space supplier. For decay, take massless positive-shell daughters summing to the rest parent $P_0=({\mathsf{M}}_0,\mathbf{0})$. With these data the identical final-state quartic total coefficient is ${\sigma }_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}=\frac{{\lambda }^2(\mathrm{\hslash }c{)}^2}{32\pi s},$ and the distinguishable-final coefficient is twice this value. The identical-daughter decay-rate coefficient is ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\mathrm{\mid }{G}_E{\mathrm{\mid }}^2}{32\pi \mathrm{\hslash }{\mathsf{M}}_0}[{\mathrm{s}}^{-1}]\mathrm{.}$ Set $\mathrm{\Gamma }:={\mathrm{\Gamma }}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}>0$. For the lifetime calculation only, let $(\mathrm{\Xi },\mathcal{F},\mathbb{P})$ be a probability space in the sense of Probability measures and probability spaces: $\mathcal{F}$ is a sigma-algebra on $\mathrm{\Xi }$, and $\mathbb{P}$ is a countably additive measure with $\mathbb{P}(\mathrm{\Xi })=1$. Let $\mathcal{B}([0,\mathrm{\infty }))$ be the sigma-algebra generated by sets $U\cap [0,\mathrm{\infty })$ for open $U\subseteq \mathbb{R}$, as in The Borel sigma-algebra of a topological space. Let $T:\mathrm{\Xi }\to [0,\mathrm{\infty })$ be a parent lifetime measured in seconds and measurable in the explicit sense that $T^{-1}(B)\in \mathcal{F}$ for every $B\in \mathcal{B}([0,\mathrm{\infty }))$. Define its law by ${\mu }_T(B):=\mathbb{P}(T^{-1}(B))$ on those Borel sets. Let $\mathrm{\ell }$ denote one-dimensional Lebesgue measure from Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, restricted to $\mathcal{B}([0,\mathrm{\infty }))$; this restriction is a measure by Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable under the stated Countable Choice assumption. Separately assume the exponential lifetime model, defined by ${\mu }_T(B)={\int }_B\mathrm{\Gamma }{e}^{-\mathrm{\Gamma }t}d\mathrm{\ell }(t)\text{for every }B\in \mathcal{B}([0,\mathrm{\infty }))\mathrm{.}$ It gives $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}(t):=\mathbb{P}(T^{-1}((t,\mathrm{\infty })))=e^{-\mathrm{\Gamma }t}$ for $t\ge 0$. Define the mean lifetime as $\mathbb{E}[T]:={\int }_{[0,\mathrm{\infty })}td{\mu }_T(t)$, and define the half-life $t_{1\mathrm{/}2}$ by $P_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{v}}(t_{1\mathrm{/}2})=1\mathrm{/}2$. Then the mean lifetime is $1\mathrm{/}\mathrm{\Gamma }$ and the half-life is $\ln 2\mathrm{/}\mathrm{\Gamma }$. The exponential law is not a consequence of the formal coefficient calculation.

The massless special case exposes both the full-sphere identical-particle factor and the distinction between a rate coefficient and a decay law.

Take $q=-e$ with $e>0$, $m>0$, and a constant field $\mathbf{B}=B_0\widehat{z}$ with $B_0>0$. It is realized by the smooth SI potential $\mathbf{A}(x,y,z)=\frac{B_0}{2}(-y,x,0),\varphi =0,$ for which $\mathrm{\nabla }\times \mathbf{A}=B_0\widehat{z}$, where $\widehat{z}=(0,0,1)$. Under the Countable Choice and adopted-action hypotheses of Minimal Dirac magnetic benchmark from the adopted action, the formal spin term is $-(q\mathrm{/}m)\mathbf{S}\cdot \mathbf{B}=(e\mathrm{/}m)S_z{B}_0$. Since ${\mathrm{\Sigma }}_z=\mathrm{diag}(1,-1,1,-1)$, its two spin eigenvalue coefficients are $\mathrm{\Delta }{E}_{\uparrow ,\downarrow }=\pm \frac{e\mathrm{\hslash }{B}_0}{2m},\mathrm{\Delta }{E}_{\mathrm{g}\mathrm{a}\mathrm{p}}=\frac{e\mathrm{\hslash }{B}_0}{m}\mathrm{.}$ Define the comparison orbital coefficient by ${\omega }_{c,\mathrm{o}\mathrm{r}\mathrm{b}}:=e{B}_0\mathrm{/}m$ and ${\nu }_{c,\mathrm{o}\mathrm{r}\mathrm{b}}:={\omega }_{c,\mathrm{o}\mathrm{r}\mathrm{b}}\mathrm{/}(2\pi )$. The cyclic coefficient frequency of the spin gap is ${\nu }_{s,\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}:=\frac{\mathrm{\Delta }{E}_{\mathrm{g}\mathrm{a}\mathrm{p}}}{2\pi \mathrm{\hslash }}=\frac{e{B}_0}{2\pi m}={\nu }_{c,\mathrm{o}\mathrm{r}\mathrm{b}}\mathrm{.}$ This is a comparison of formal minimal coefficients. It is not the full Landau spectrum, a radiative correction, a prediction of zero measured anomaly, or a laboratory measurement.

The uniform field makes the spin eigenvalues and the formal coefficient comparison explicit.